How do they manage that? Do you have to visit a government office to get a passport? I only have a UK passport and it definitely does not include my finger prints as it was obtained by post.

Yes.

In fact, you have to do it twice; once to apply and once to receive it.

In Poland at least, the queues are often a few weeks long, and you have to book online in advance. If it's summer season and you need to travel soon, you usually have to go to some office in Buttfuckville at the other end of the country to get a reasonable date. Passport office hunting is an established practice for people who need one ASAP.

If you're a kid, both of your parents need to be present with you at the appointment at the same time, so they have to coordinate time off work. The fun begins if they're divorced and one lives in a different country.

At least, unlike in the UK/US, the process itself is pretty painless. I got mine just after I turned 18 (see above), and all I really needed to do was to go there, bring my ID, fill a single form and give them the fingerprints.